Dr. Byrne Math 125

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4.3/4.4 Comparing The First and Second Derivative Tests Home Work: Turn this in for a Quiz Grade

Definition of a critical point: A number c in the domain of f (x) is called a critical point if f '(c) = 0 or f '(c) does not exist.

Comments:  The critical points of f (x) are always defined by the zeros (or undefined points) of the first derivative.  In order to be a critical point, a point must be in the domain of f (x). Often points for which f '(x) is not defined are also points where f (x) is not defined, so we would exclude these.

PROS AND CONS OF THE TWO TESTS: The first and second derivative tests can be used to identify a critical point as a minimum or a maximum. The first derivative requires testing the sign of TWO points around the critical point, but always provides a classification. The second derivative test requires testing the sign of ONE point around the critical point, but is sometimes inconclusive.

1st Derivative Test

Let f (x) be differentiable in a neighborhood containing c. If c is a critical point, then

 if 푓′(푥) changes from + to − at c, 푓(푐) is a local maximum.  if 푓′(푥) changes from − to + at c, 푓(푐) is a local minimum.  if 푓′(푥) does not change sign at c, 푓(푐) is neither a minimum nor a maximum.

2nd Derivative Test

Let f (x) be twice-differentiable in a neighborhood containing c. If c is a critical point, then

 if 푓′′(푐) > 0, 푓(푐) is a local minimum  if 푓′′(푐) < 0, 푓(푐)is a local maximum  if 푓′′(푐) = 0 or does not exist, the test is inconclusive (푓(푐) may be local max, local min or neither) (Note: An inflection point occurs where 푓′′(푥) exists and the concavity changes.) Example A: 푦 = 3푥 − 푥3

1) On the graph provided, label all maxima, minima and inflection points (max/min/I).

2) Find all critical points of y:

3) First derivative test:

a. Identify the intervals where y is increasing or decreasing:

y increasing: . y decreasing: .

b. What does the first derivative test tell us about each critical point? (Review how the test works on the first page of this worksheet.)

4) Second derivative test:

a. Identify the intervals where y is concave up or concave down:

y concave up: . y concave down: .

b. What does the second derivative test tell us about each critical point?

Example B: 푦 = 푥5 − 5푥4

1) On the graph provided, label all maxima, minima and inflection points (max/min/I).

2) Find all critical points of y:

3) First derivative test:

a. Identify the intervals where y is increasing or decreasing:

y increasing: . y decreasing: . .

b. What does the first derivative test tell us about each critical point?

4) Second derivative test:

a. Identify the intervals where y is concave up or concave down:

y concave up: . y concave down: . .

b. What does the second derivative test tell us about each critical point?

Example C: 푦 = 5푥2/5 − 2푥

1) On the graph provided, label all maxima, minima and inflection points (max/min/I).

2) Find all critical points of y:

3) First derivative test:

a. Identify the intervals where y is increasing or decreasing:

y increasing: . y decreasing: . .

b. What does the first derivative test tell us about each critical point?

4) Second derivative test:

a. Identify the intervals where y is concave up or concave down:

y concave up: . y concave down:

b. What does the second derivative test tell us about each critical point?

Example D: Comparing the 1st and 2nd derivative tests for classifying the critical point of 푓(푥) = 푥3 + 3.

(a) Find the critical point of f (x).

(b) What does the first derivative test tell us about this critical point?

(c) What does the second derivative test tell us about this critical point?

Example E: st nd ( ) 1 4 Comparing the 1 and 2 derivative tests for classifying the critical point of 푓 푥 = 4푥 + 1.

(a) Find the critical point of f (x).

(b) What does the first derivative test tell us about this critical point?

(c) What does the second derivative test tell us about this critical point?

Example F: Comparing the 1st and 2nd derivative tests for classifying the critical point of 푓(푥) = 푥(푥 − 1).

(a) Find any critical points of f (x).

(b) What does the first derivative test tell us about each critical point?

(c) What does the second derivative test tell us about each critical point?

Example G: 4 Comparing the 1st and 2nd derivative tests for classifying the critical point of 푓(푥) = 1 − . 푥2

(a) Find any critical points of f (x).

(b) What does the first derivative test tell us about each critical point?

(c) What does the second derivative test tell us about each critical point?